On the numerical simulation of particle dynamics in the radiation belt: 1. Implicit and semi-implicit schemes

The particle dynamics in the Earth's radiation belt is generally modeled by means of a two-dimensional diffusion equation for the particle distribution function in energy and pitch angle. The goal of this paper is to survey and compare different numerical schemes for the solution of the diffusion equation, and to outline the optimal strategy from a numerical point of view. We focus on the general (and more computationally challenging) case where the mixed terms in the diffusion tensor are retained. In Part 1, we compare fully implicit and semi-implicit schemes. For the former, we have analyzed a direct solver based on a LU decomposition routine for sparse matrices, and an iterative incomplete LU preconditioned Generalized Minimal REsidual solver. For the semi-implicit scheme, we have studied an alternating direction implicit scheme. We present a convergence study for a realistic case that shows that the time step and grid size are strongly constrained by the desired accuracy of the solution. We show that the fully implicit scheme is to be preferred in most cases as the more computationally efficient.